Hang Thi Anh Nguyen, Michael Hoff, Truong Le Hoang
We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.
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We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $G_a$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We consider extremal contractions on smooth Fano fourfolds whose second Chern character is positive. We show that such contractions can neither be of fiber type nor contract a divisor to a point.
On locally factorial Fano fourfolds of Picard number two
本文分类了具有超曲面Cox环且允许三维环面有效作用的Picard数为2的局部析因Fano四重折叠。
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff ective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in our papers jointly with Takashi Kishimoto.
We prove that certain Fano fourfolds of K3 type constructed by Fatighenti-Mongardi have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.
In this paper we show that a general element of $|-K_X|$ on a four-dimensional Fano manifold has at most terminal singularities. We then determine an explicit local expression of these singular points.
Fano fourfolds of K3 type
本研究从旗流形乘积上齐次向量丛零轨迹构建的法诺四簇数据库中提取64个K3型族，探究其几何与K3结构起源，给出不变量、有理性及导出范畴半正交分解相关信息。
Prime Fano fourfolds in classical and generalized Grassmannians
本文在广义Grassmannian中完整分类了Picard秩1指数1的Fano四重簇，扩展了Küchle的结果，发现了三个新族，并计算了体积和Hodge数等不变量。
The affine cones over Fano-Mukai fourfold of genus $7$ are flexible
本文证明了亏格 $7$ 的任何光滑 Fano-Mukai 四重折叠的仿射锥具有柔性。
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
On smooth Fano fourfolds of Picard number two
本文分类了具有一般超曲面Cox环且Picard数为2的光滑Fano四维簇。
We show that the affine cones over any Fano-Mukai fourfold of genus 10 are flexible; in particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano-Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.
Derived categories of Gushel-Mukai surfaces and Fano fourfolds of K3 type
研究K3型Fano四维流形的半正交分解，证明非常一般的对偶Gushel-Mukai表面不相等，尽管导出和L等价。
We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors on birational models obtained by sequences of flips.The results presented here are part of the author's Ph.D. thesis, written under the supervision of Stefan Kebekus.
We prove that the Fano variety of lines of a cuspidal cyclic cubic fourfold is a symplectic variety with transversal A2-singularities and we study the properties of the nonsymplectic order three automorphism induced by the covering automorphism on the irreducible holomorphic symplectic manifold obtained by blowing up the singular locus.
Classification of Fano fourfolds with large anticanonical base locus
本文分类了反典范基轨迹为光滑曲面的光滑Fano四重，证明恰有22个形变族，均源于同一几何构造，并与具有$\mathbb P^1$-丛结构的Fano三重相关。
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
In this text we prove that if a smooth cubic in $\PR^5$ has its Fano variety of lines birational to the Hilbert scheme of two points on a K3 surface, then there exists a smooth projective curve or a smooth projective surface embedded in the Fano variety, such that the kernel of the push-forward (at the level of zero cycles ) induced by the closed embedding is torsion.
We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families $\mathcal C_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.
Terminalizations of quotients of Fano varieties of lines on cubic fourfolds
本文研究了立方四维流形上线的Fano簇商的终端化，计算了相关拓扑不变量，并识别出新的四维不可约解析辛流形变形类。
Cubic fourfolds with birational Fano varieties of lines
本文给出了几对非同构三次四重体例子，证明其 Fano 直线簇是有理等价的，为 Fourier-Mukai 伙伴猜想提供新证据，并提出三次四重体与其 Fano 直线簇有理等价性相关的猜想。
This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth $4$-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi-Yau-$5$ algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.
Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose class [C] belongs to R. We show that s(X)\leq n if n\leq 4. And for n\leq 4, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.
On the geometry of some quiver zero loci Fano fourfolds (with an appendix by E. Kalashnikov and F. Tufo)
本文对 Kalashnikov 提出的 170 族箭图旗零轨迹 Fano 四维流形进行研究，将其视为齐性向量丛截面的零轨迹，并给出全部 170 族的双有理与双正则刻画。
The Fano variety of lines on singular cyclic cubic fourfolds
本文研究奇异循环三次四重簇上直线Fano簇的辛解消，特别关注沿具有$A_i$型孤立奇点的三次三重簇分支的情形，并发现覆盖自同构诱导的三阶非辛自同构。
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2,5). Special fourfolds correspond to a countable union of hypersurfaces in the …
An irregular smooth Fano fourfold in positive characteristic
在特征二的代数闭域上构造了一个光滑射影 Fano 四维簇，具有非零的 $H^1$ 且无自由有理曲线，从而不是可分有理连通的，该例子借助 ChatGPT 构建。
Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls
研究包含三次滚动面对的三次四重线上Fano簇（超Kähler四重）的双有理几何，识别所有双有理模型并给出显式映射，同时刻画双有理自同构群的生成元和关系。
The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 with the automorphism groups $GL_2(k)\rtimes\mathbb{Z}/2\mathbb{Z}$ and $(\mathbb{G}_a\times\mathbb{G}m)\rtimes\mathbb{Z}/2\mathbb{Z}$, respectively. For any other Fano-Mukai fourfold $V$ of genus 10 one has $\mathrm{Aut}(V)=\mathbb{G}_m^2\rtimes \math…
Constant cycle surfaces on Fano varieties of cubic fourfolds
本文研究光滑四次三维簇的线Fano簇上常循环曲面的有限性，证明了固定阶的形如F(Y)的常循环曲面至多有限个。
Central Weyl involutions on Fano-Mukai fourfolds of genus 10
本文研究了亏格10的Fano-Mukai四重流形上源自G2型Weyl群中心的唯一对合τ，详细刻画了其不动点集合及由τ不变直线扫出的直纹面。
Fano fourfolds with large anticanonical base locus
针对法诺流形反典范除子的奇异性问题，本文对比三元体与四元体的差异，证明当四维法诺流形反典范系统基底轨迹为余维二的正规曲面时，所有反典范除子均奇异。
Lines on cubic threefolds and fourfolds containing a plane
该研究刻画了特征非2域上含平面的一般三次三维簇的直线Fano概形，以此刻画非闭域上这类三次三维簇的有理性，并显式构造含平面三次四维簇直线Fano簇的拉格朗日纤维化。
On a numerical criterion for Fano fourfolds
本文证明了Campana--Peternell猜想在四维情形的一个特例，即满足特定数值条件且反典范除子严格数值有效的射影光滑四折叠必为Fano四折叠，并完全解决了相关非消失猜想。
Cohomological rigidity of smooth toric Fano fourfolds
该研究证明整上同调环分次同构的光滑环面法诺四维簇必微分同胚，确立其微分同胚类由上同调环决定的刚性性质。
K{ü}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.
Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing
本文证明当前已知的 Picard 秩 1 指数 1 的光滑 Fano 四重族满足 χ(X,T_X)<0，结合指数>1 情形，表明除射影空间外所有已知 Picard 秩 1 的光滑 Fano 四重均不满足 Bott 消失，从而任何具有大于 1 次自同态的此类簇必为 P^4。
CscK metrics on rank one spherical Fano fourfolds
本文探索低维低秩球面簇任意 Kähler 类的显式 K-stability，将秩一球面簇的组合判据应用于射影平面乘积沿对角线的吹胀，强烈表明其每个 Kähler 类均存在 cscK 度量。
Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds
本文证明终端和典范 Gorenstein Fano 四折叠的反典范体积上界，并给出尖锐界及特殊情形下的改进。
It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}…
The Fano of lines, the Kuznetsov component, and a flop
本文证明了三次超曲面导出范畴的Kuznetsov分量的对称平方等价于直线Fano簇的导出范畴，验证了Galkin猜想，其关键步骤是建立了一个特定12维flop的新导出等价。
In this paper, we prove a general principle of lifting an automorphism from positive characteristic to zero characteristic. We based on the principle to prove the automorphism group of Fano variety of cubic threefold (fourfold) acts on its cohomology group faithfully. We also prove the same result for a singular cubic threefold (fourfold) with one ordinary double singularity. As an application, we classify the automorphism groups of cubic fourfolds with small middle Picard number over complex numbers.
We give an interpretation of the Fano variety of lines on a cubic fourfold and of the hyperkahler eightfold, constructed by Lehn, Lehn, Sorger and van Straten from twisted cubic curves in a cubic fourfold non containing a plane, as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a consequence, we reprove the categorical version of Torelli Theorem for cubic fourfolds, we obtain the identification of the period point of LLSvS eightfold with that of the Fano variety, and we discuss derived Torelli Theorem for cubic fourfolds.
On the Chow Ring of Fano Fourfolds of K3 type
本文证明了一类新构造的 K3 型 Fano 簇具有乘性 Chow-Künneth 分解，使其 Chow 环表现如 K3 曲面，并给出了爆破射影簇 Franchetta 性质的判据。
Irrational families of Fano weighted complete intersection fourfolds
该研究基于格罗莫夫-维滕不变量的原子论方法，证明了加权射影空间中广泛族光滑法诺完全交四元重流形的辛非有理性，填补了此前仅对一般成员成立的结论空白。
Amplified endomorphisms of Fano fourfolds
本文证明具有圆锥曲线丛结构的光滑 Fano 四维簇为环簇当且仅当它允许放大自同态，此时该簇为有理簇。
O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.
We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.
In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.
Exceptional Collections for Toric Fano Fourfolds
本文推广了对角分解的相关结果，研究了四维光滑射影环面法诺簇的例外系，发现124个簇中有72个可通过 Hanlon-Hicks-Lazarev 对角分解得到完全强线丛例外系，且与 Bondal 数值判据等价。
We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first dealt with by Kovalev in \cite{K03}. Here in this paper an admissible pair $(\overline{X},D)$ consists of a compact Kähler manifold $\overline{X}$ and a smooth anticanonical divisor $D$ on $\overline{X}$. If two admissible pairs $(\overline{X}_1,D_1)$ and $(\overline{X}_2,D_2)$ wi…
Even nodal surfaces of K3 type
本文研究具有二次曲线丛结构的K3型Fano四重态，构造其与hyperKähler簇的几何联系，描述了作为Kummer四次曲面推广的节点曲面族。
We show that the affine cones over a general Fano-Mukai fourfold of genus $g=7$, $8$ and $9$ are flexible. Equivalently, there is an infinitely transitive action of the special automorphism group on such affine cones. In particular, any Mukai fourfold of genus $7,8$ and $9$ is $\Bbb A^2$-cylindrical.
We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture.
On Arithmetic Mirror Symmetry for smooth Fano fourfolds
本文引入了一类显式的调和Laurent多项式（Villegas和Doran-Kerr意义下），包含所有具有极丰反典范类的光滑Fano三维流形的Landau-Ginzburg模型，并验证了其包含多种Fano四维流形（如光滑环簇和Grassmannian中的完全交、quiver旗零轨迹）的Landau-Ginzburg模型，进而讨论了算术镜像对称猜想以及Golyshev-Kerr-Sasaki提出的基于Hodge理论的Fano流形Apéry常数研究，最后利用Kerr证明的算术镜像对称猜想特例构造了两个具体代数类之间的镜像对称对应。
On the Fano dimension of an Enriques surface
本文通过构造Fano四维流形，将一般Enriques曲面的Fano维度从六降低到四。
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